A subset H of a Euclidean space X is said to be compact (in X) if every open covering of H has a finite subcovering.Originally Posted by Panconleche
Here are a few examples.
Consider the interval can and the cover
This covers [1,2] for all values of n but it also covers for any finite value of n.
Now consider (1,2) and the cover above this covers it, but it must be true for every covering.
at "infinity" it covers (1,2) but doesn't for any finite value of n.
so (1,2) is not compact
I hope this helps.